7/20 



THE ULTRA STABLE SYSTEM 



break if its variables are driven far enough away from their usual 

 values. Thus, machines with moving parts, if driven ever faster, 

 will break mechanically; electrical apparatus, if subjected to 

 ever higher voltages or currents, will break in insulation ; machines 

 made too hot will melt — if made too cold they may encounter 

 other sudden changes, such as the condensation which stops a 

 steam-engine from working below 100° C; in chemical dynamics, 

 increasing concentrations may meet saturation, or may cause 

 precipitation of proteins. 



Although there is no rigorous law, there is nevertheless a wide- 

 spread tendency for systems to show changes of step-function 

 form if their variables are driven far from some usual value. 

 Later (S. 9/7) it will be suggested that the nervous system is not 

 exceptional in this respect. 



Systems containing step-mechanisms 



7/20. When a state-determined system includes a step-function 

 among its variables, the whole behaviour can undergo a simplifica- 

 tion not possible when the variables are all full-functions. 



Suppose that we have a system with three variables, A, B, S; 

 that it has been tested and found state-determined; that A and 

 B are full-functions ; and that S is a step-mechanism. (Variables 

 A and B, as in S. 21/7, will be referred to as main variables.) 

 The phase-space of this system will resemble that of Figure 7/20/1 

 (a possible field has been sketched in). The phase-space no longer 

 fills all three dimensions, but as S can take only discrete values, 

 here assumed for simplicity to be a pair, the phase-space is 

 restricted to two planes normal to S, each plane corresponding to a 



Figure 7/20/1 : Field of a state-determined system of three variables, of 

 which S is a step-function. The states from C to C are the critical states 

 of the step-function for lines in the lower plane. 



93 



